In aided diagnosis of early Alzheimer disease (AD), it is important to identify features of the ROIs (reactive oxygen intermediates including entorhinal cortex and hippocampus) in the magnetic resonance imaging (MRI) images. However, the MRI technology can only use the hippocampus atrophy as one of the indicators to distinguish a patient from a healthy person. The explanations of doctors to the MRI images are easily affected by subjectivity and lack of consistency. Moreover, it is hard for the MRI technology to evaluate correctly how serious the symptom of the AD is.
1, the Prior Image Processing Technology
Contourlet Transform
The Contourlet transform inherits the anisotropic scale relationship of Curvelet transform. In a certain sense, the Contourlet transform is another implementation of the Curvelet transform. The basic idea of Contourlet transform is to capture edge point discontinuities using multiscale decomposition similar to wavelet firstly, and then to converge the point discontinuities with similar location to form contour segments according to direction information.
The Contourlet transform may be divided into two parts: Laplacian Pyramid (LP) filter structure and 2-D directional filter bank (DFB). The LP decomposition first produce a lowpass sampling approach of the original signals and a difference image of the original image and a lowpass predication image, and further decompose the lowpass image obtained to gain a lowpass image and a difference image of the next level. Multiresolution decomposition of image is obtained in this gradually filtering. The DFB adopts conjugate mirror filters with a sector structure, and transforms one level binary tree structure of a directional filter into a structure with 21 parallel passages.
The Contourlet transform is a new two-dimensional transform method for image representations, which has properties of multiresolution, localization, directionality, critical sampling and anisotropy. Its basic functions are multiscale and multidimensional. The contours of original images, which are the dominant features in natural images, can be captured effectively with a few coefficients by using Contourlet transform.
However, when the new transform method is used to process MRI images of different body parts, it needs to reconstruct new algorithm and select suitable parameters using basis functions. Therefore, there still are many theoretical issues to be studied. The Contourlet transform has been successfully used to solve actual problems such as image fusion. However, there are few references on extraction of image texture features using the Contourlet transform. No one has disclosed to use the second generation wavelet transform and Contourlet transform to extract textures of AD brain MRI images.
2, the Prior Typical Prediction Model Includes Supervised Machine Learning Model and Semi-Supervised Machine Learning Model
Supervised Machine Learning Model
Gaussian Process (GP):
The Gaussian process classification algorithm is a supervised learning algorithm, which is established under the Bayesian framework, and it is first used to solve the problem of nonlinear value prediction. Provided there is a data set D={(xi, yi)|yi=±1, i=1, 2, . . . , n}, the essence of binary classification is to find a suitable mapping f(x) to ensure that the sample can be classified correctly using y=f(x). The basic idea of classification using Gaussian regression algorithm is to find a suitable f(x) aiming at maximizing the posterior probability in Bayesian rule under the assumption that the f(x) is a Gaussian process. It is firstly assumed that there is an implicit function f(x)=Ø(x)Tw, which defines the mapping relationship between input properties and output properties (class tags). It is further assumed that the class tags are independent identically distributed, and are subjected to conditional probability p(y|f(x)). Thus, for the binary classification, the probability that the sample xi belongs to class yi is:p(yi|fi)=σ(yifi)
wherein, fi=f(xi) is an implicit function, σ(·) is a Sigmoid class function, such as a logistic function or a cumulative Gaussian function. Since training samples are independent to each other, the corresponding likelihood probability is:
      p    ⁡          (              y        |        f            )        =                    ∏                  i          =          1                n            ⁢                          ⁢              p        ⁡                  (                                    y              i                        |                          f              ⁡                              (                                  x                  i                                )                                              )                      =                  ∏                  i          =          1                n            ⁢                          ⁢              σ        ⁡                  (                                    y              i                        |                          f              i                                )                    
wherein, for the sample set D={(xi, yi)|yi=±1, i=1, 2, . . . , n}, the Gaussian process classifier assumes that the f(x)˜GP(0, k) is a Gaussian process with mean value equivalent to zero to simplify calculation, i.e. p(f|x) is a multidimensional Gaussian density function:
      p    ⁡          (              f        |        x            )        =            1                                    (                          2              ⁢              π                        )                                L            ⁢                          /                        ⁢            2                          |        k        ⁢                  |                      1            ⁢                          /                        ⁢            2                                ⁢    exp    ⁢          {                        -                      1            2                          ⁢                  f          T                ⁢                  K                      -            1                          ⁢        f            }      
wherein, K is a covariance matrix, which is a kernel function. Therefore, the posterior probability can be obtained according to Bayesian formula:
      p    ⁡          (                        f          |          y                ,        X            )        =                    p        ⁡                  (                      y            |            f                    )                    ⁢              p        ⁡                  (                      f            |            X                    )                            p      ⁡              (                  y          |          X                )            
wherein, p(y|f) is a likelihood function, p(y|X)=∫p(y|f)p(f|x)df is a marginal probability density.
The training to the Gaussian process classifier is namely the estimation to the posterior probability. The estimated value q(f|y, X) of the posterior probability p(f|y,X) can be calculated by Laplace method. Therefore, the probability that the class tag of a given testing data point x* belongs to positive class is:π*=p(y*=±1|X,y,x*)=∫Ø(f*)q(f*|X,y,x*)df* 
According to the principle of Gaussian process classification, the numeric data set obtained after feature extraction is inputted into the Gaussian process classifier. The estimated value q(f|y, X) of the posterior probability can be calculated after hyper-parameter optimization. Then, prediction results related to testing set can be obtained.
1) Support Vector Machine (SVM):
Support vector machine method is a machine learning method based on VC dimension theory of statistical learning theory and structure risk minimum principle. Its principle is to find an optimal classification hyperplane which satisfies the requirements of classification, so that the hyperplane ensures classification accuracy and the margin of the hyperplane is largest.
In theory, the support vector machine is capable of realizing the optimal classification of linear classifiable data. Taking the classification of two kinds of data as an example, it is provided that the training sample set is (xi, yi), i=1, 2, . . . , l, xε{±+1}, and the hyperplane is written as (w·xi)+b=0. In order to ensure that the classification plane is correct to all the samples and has a classification space, the following restriction should be satisfied: yi[(w·xi)+b]≧1, i=1, 2, . . . , l. It can be calculated that the classification space is 2/∥w∥. Thus, the problem of constructing an optimal hyperplane has been transformed to the problem of solving the following formula under the restriction formula:minØ(w)=½∥w∥2=½(w·w)
In order to solve this problem of restriction optimization, the Lagrange function is introduced:L(w,b,a)=½∥w∥−α(y(w·x)+b)−1
wherein, α>0, and it is a Lagrange multiplier. The solution of the problem of restriction optimization is determined by the saddle point of the Lagrange function. Moreover, the partial derivative of the solution of the problem of restriction optimization at the saddle point with respect to w and b is zero. Thus, the QP (quadratic programming) problem is transformed to a corresponding dual problem:
            max      ⁢                          ⁢              Q        ⁡                  (          α          )                      =                            ∑                      j            =            1                    l                ⁢                                  ⁢                  α          j                    -                        1          2                ⁢                              ∑                          i              =              1                        l                    ⁢                                          ⁢                                    ∑                              j                =                1                            l                        ⁢                                                  ⁢                                          α                i                            ⁢                              α                j                            ⁢                              y                i                            ⁢                                                y                  j                                ⁡                                  (                                                            x                      i                                        ·                                          x                      j                                                        )                                                                                            s        .        t        .                                  ⁢                              ∑                          j              =              1                        l                    ⁢                                          ⁢                                    α              j                        ⁢                          y              j                                          =      0        ,          j      =      1        ,    2    ,    …    ⁢                  ,    l  
the optimal solution α*=(α1*, α2*, . . . , αl*)l is obtained.
Calculating the optimal weight vector w* and optimal bias b*, which are:
            w      *        =                  ∑                  j          =          1                l            ⁢                          ⁢                        α          j          *                ⁢                  y          j                ⁢                  x          j                                b      *        =                  y        i            -                        ∑                      j            =            1                    l                ⁢                                  ⁢                              y            j                    ⁢                                    α              j              *                        ⁡                          (                                                x                  j                                ·                                  x                  i                                            )                                          
Wherein, the subscript jε{j|αj*>0)}. Thus, the optimal classification hyperplane is obtained: (w*·x)+b*=0, and the optimal classification function is:
            f      ⁡              (        x        )              =                  sgn        ⁢                  {                                    (                                                w                  *                                ·                x                            )                        +                          b              *                                }                    =              sgn        ⁢                  {                                                    ∑                                  j                  =                  1                                l                            ⁢                                                          ⁢                                                α                  j                  *                                ⁢                                                      y                    j                                    ⁡                                      (                                                                  x                        j                                            ·                                              x                        i                                                              )                                                                        +                          b              *                                }                      ,      x    ∈          R      n      
For the linear unclassifiable case, the main idea of SVM is to map the input vector to a high dimensional eigenvector space, and construct the optimal classification plane in the feature space.
Doing transform Ø to x from the input space Rn to the feature space H, it can be obtained:x→Ø(x)=(Ø1(x),Ø2(x), . . . Øl(x))T 
Replacing the input vector x with the eigenvector Ø(x), the optimal classification function can be obtained:
      f    ⁡          (      x      )        =            sgn      ⁡              (                              w            ·                          ∅              ⁡                              (                x                )                                              +          b                )              =          sgn      ⁡              (                                            ∑                              j                =                1                            l                        ⁢                                                  ⁢                                          α                i                            ⁢                              y                i                            ⁢                              ∅                ⁡                                  (                  x                  )                                                              +          b                )            
2) Random Forests (RF):
Random forest is a classification and predication model proposed by Leo Breiman, which has been applied in many fields.
The basic idea of constructing random forests is continuously generating training sample and testing sample by the technique of bootstrap resampling. The training sample generates a plurality of classification tree constituting the random forests. The classification result of the testing data is determined according to the vote of classification tree.
It is provided that the content of original sample is N, the variables representing various genes are x1, x2, . . . , xm. Randomly sampling with replacement new self-service samples with the number of b, and hence forming classification trees with the number of b. The samples that are not sampled each time form the data out of bag (OOB) with the number of b. the data out of bag acting as testing sample can be used to evaluate the importance of the various variables in the classification. The realization process is as follows:
Using the self-service samples to form each tree classifier, and classifying the corresponding OOB at the same time, it can hence obtain the voting scores of each sample in the OOB of the self-service samples with the number of b, which can be written as vote1, vote2, . . . , votes.
Randomly changing the order of the value of the variables xi in the OOB samples with the number of b, it forms new OOB testing samples. Then, classifying the new OOB by the constructed random forests and obtaining the voting scores of each sample according to the number of sample that is judged to be correct.
3) Lasso:
The basic idea of Lasso (the least absolute shrinkage and selection operator) method is to minimize the sum of squared residuals under the restriction that the sum of absolute regression coefficient is less than a constant, thus generating some regression coefficient rigidly equaling to zero. Although this is a biased estimation to decrease the variance of the predicted value, the prediction accuracy of the model will be increased accordingly and the model obtained is easier to be explained. For the case with a plurality of independent variables, we always hope to determine a model with fewer variables to present the best effect. The selection criteria of Lasso is to minimize the penalty function
                    1        n            ⁢                        ∑                      i            =            1                    n                ⁢                                  ⁢                              (                                          Y                i                            -                                                X                  i                                ⁢                                  β                  i                                                      )                    2                      +                  ∑                  j          =          1                λ            ⁢                          ⁢                                β          j                              ,and adopts the least-angle regression algorithm.
The regression steps of Lasso are as follows:
a) Provided the estimated initial value of β is β(0), fixing λ and selecting the least-squares estimation {tilde over (β)}LS as the initial value of β. That is to say, β(0)={tilde over (β)}LS=(X′X)−1X′Y;
b) Obtaining the estimation to β(written as {tilde over (β)}) when fixing λ according to the estimated initial value β(0) and the following iterative formula:β(1))=β(0)−[∇2QLSβ(0)+nΣλβ(0)]−1[∇QLSβ(0)+nUλβ(0)]
wherein, ∇2QLSβ(0)=X′X, ∇QLSβ(0)=−2X(Y−Xβ(0)), Σλβ(0)=λdiag(1/|β0|, . . . 1/|βp|), if βj=0, the value of 1/|βj| can be evaluated to be a very large number. Uλβ(0)=Uλβ(0)β(0);                c) Changing the value of λ in the range of 0˜1, under the restriction of λ, constructing generalized cross validation        
      GCV    ⁡          (      λ      )        =                                                    Y            -                          X              ⁢                                                          ⁢                                                β                  ~                                λ                                                              2                              (                      1            -                                          e                ⁡                                  (                  λ                  )                                            /              n                                )                2              .  The penalty function that minimizes the cross validation is the optimal selection penalty parameter λ, λ=arg min GCV(λ), and hence the Lasso estimation to β is {circumflex over (β)}Lasso={circumflex over (β)}Lasso.                (1) Semi-Supervised Support Vector Machine (S3VM):        
The semi-supervised learning is a key problem studied in the field of pattern recognition and machine learning in recent years, which mainly considers the problem of how to train and classify with a few tag samples and a plurality of samples without tag. The semi-supervised learning algorithm based on support vector machine is a model first proposed by Kristin, which is realized by mixed integer programming method. Later, Joachims, Demirez et al. proposed some similar methods one after another among which the transductive support vector machine (TSVM) algorithm is more typical.
Given that a training set T={(x1, y1), . . . , (xl, yl), xj+1, . . . , xj+k}, wherein, xfεX=Rn, the top l samples are samples with classification tag, i.e. when i=1, 2, . . . , l, yiεY={−1,1}; the down k samples are samples without classification tag. Finding a decision function f(x)=sgn(ωTφ(x)+b) at X=Rn to deduce the class (positive class or negative class) of x of any pattern (wherein, ω is weight vector, φ(·) is a mapping function, b is a constant, and the kernel function K(xi, xj)=<φ(xi), φ(xj)>). Accordingly, the solving process is actually to find a principle to divide the points in Rn into two parts.
Obviously, it exists ωεRn, bεR, to ensure that it exists yi(ωTφ(xi)+b)+τi≧1, ξi≧0, i=1, . . . , l for each sample with classification tag xi(i=1, . . . , l). It exists (ωTφ(xj)+b)+rj≧1, ωTφ(xj)+b−sj≦−1, rj, sj≧0, j=l+1, . . . , l+k.
Currently, such a classification learning machine is constructed by maximizing soft margin and slack variables ξ=(ξ1, ξ2, . . . , ξl), r=(rl+1, rl+2, . . . , rl+k), s=(sl+1, sl+1, . . . , sl+k) are introduced. Obviously, ξ, r, s reflect the case that the training set is wrongly classified. Thus, it can construct the degree that describes the training set wrongly classified, and it may as well to adopt Σi=1lξi2, Σj=1l+krjsi as a mensuration to degree that describes the training set wrongly classified. At the same time, it is expected that the wrongly classified degree Σi=1lξi2, Σj=1l+krjsi is as small as better, that the optimal classified hyperplane can be solved out by minimizing ½(ωTω+b2), and that the affection to the classification result is limited. In order to fully take the three aims into account, the penalty parameters c1 and c2 are introduced as a weight to balance the three aims. That is to say, the aim function is minimized as:
            1      2        ⁢          (                                    ω            T                    ⁢          ω                +                  b          2                    )        +            c      1        ⁢                  ∑                  i          =          1                l            ⁢                          ⁢              ξ        i        2              +            c      2        ⁢                  ∑                  j          =          1                          l          +          k                    ⁢                          ⁢                        r          j                ⁢                  s          i                    
Thus, the problem of solving the optimal classified hyperplane is transformed into the problem of optimization:
                    ⁢                  min                  ω          ,          b          ,          ξ          ,          r          ,          s                    ⁢              (                                            1              2                        ⁢                          (                                                                    ω                    T                                    ⁢                  ω                                +                                  b                  2                                            )                                +                                    c              1                        ⁢                                          ∑                                  i                  =                  1                                l                            ⁢                                                          ⁢                              ξ                i                2                                              +                                    c              2                        ⁢                                          ∑                                  j                  =                  1                                                  l                  +                  k                                            ⁢                                                          ⁢                                                r                  j                                ⁢                                  s                  i                                                                    )                                ⁢                                        s            .            t            .                                                  ⁢                                          y                i                            ⁡                              (                                                                            ω                      T                                        ⁢                                          φ                      ⁡                                              (                                                  x                          i                                                )                                                                              +                  b                                )                                              +                      ξ            i                          ≥        1            ,              i        =        1            ,      …      ⁢                          ,      l                          ⁢                                                      ω              T                        ⁢                          φ              ⁡                              (                                  x                  j                                )                                              +          b          +                      r            j                          ≥        1            ,                                                  ω              T                        ⁢                          φ              ⁡                              (                                  x                  j                                )                                              +          b          -                      s            j                          ≤                  -          1                    ,              r        j            ,                        s          j                ≥        0            ,                          ⁢                          ⁢              j        =                  l          +          1                    ,      …      ⁢                          ,              l        +        k                                ⁢    Given              L      ⁡              (                  ω          ,          b          ,          ξ          ,          r          ,          α          ,          β          ,          γ          ,          u          ,          v                )              =                            1          2                ⁢                  (                                                    ω                T                            ⁢              ω                        +                          b              2                                )                    +                        c          1                ⁢                              ∑                          i              =              1                        l                    ⁢                                          ⁢                      ξ            i            2                              +                        c          2                ⁢                              ∑                          j              =              1                                      l              +              k                                ⁢                                          ⁢                                    r              j                        ⁢                          s              i                                          -                        ∑                      i            =            1                    l                ⁢                                  ⁢                              α            i                    ⁡                      [                                                            y                  i                                ⁡                                  (                                                                                    ω                        T                                            ⁢                                              φ                        ⁡                                                  (                                                      x                            i                                                    )                                                                                      +                    b                                    )                                            +                              ξ                i                            -              1                        ]                              -                        ∑                      j            =                          l              +              1                                            l            +            k                          ⁢                                  ⁢                              β            j                    ⁡                      (                                                            ω                  T                                ⁢                                  φ                  ⁡                                      (                                          x                      j                                        )                                                              +              b              +                              r                j                            -              1                        )                              -                        ∑                      j            =                          l              +              1                                            l            +            k                          ⁢                                  ⁢                              γ            j                    ⁡                      (                                                            -                                      ω                    T                                                  ⁢                                  φ                  ⁡                                      (                                          x                      j                                        )                                                              -              b              +                              r                j                            -              1                        )                              -                        ∑                      j            =                          l              +              1                                            l            +            k                          ⁢                                  ⁢                              u            j                    ⁢                      r            j                              -                        ∑                      j            =                          l              +              1                                            l            +            k                          ⁢                                  ⁢                              v            j                    ⁢                      s            j                                                  ⁢          Thus      ,                                    ∂            L                                ∂            ω                          =                  0          =                                    >              ω                        =                                                            ∑                                      i                    =                    1                                    l                                ⁢                                                                  ⁢                                                      y                    i                                    ⁢                                      α                    i                                    ⁢                                      φ                    ⁡                                          (                                              x                        i                                            )                                                                                  +                                                ∑                                      j                    =                                          l                      +                      1                                                                            l                    +                    k                                                  ⁢                                                                  ⁢                                                      (                                                                  β                        j                                            -                                              γ                        j                                                              )                                    ⁢                                      φ                    ⁡                                          (                                              x                        j                                            )                                                                                                              ,                          ⁢                          ⁢                                    ∂            L                                ∂            b                          =                  0          =                                    >              b                        =                                                            ∑                                      i                    =                    1                                    l                                ⁢                                                                  ⁢                                                      y                    i                                    ⁢                                      α                    i                                                              +                                                ∑                                      j                    =                                          l                      +                      1                                                                            l                    +                    k                                                  ⁢                                                                  ⁢                                  β                  j                                            -                              γ                j                                                        ,                          ⁢                          ⁢                                    ∂            L                                ∂            b                          =                  0          =                                    >                              ξ                i                                      =                                          1                                  2                  ⁢                                                                          ⁢                                      c                    1                                                              ⁢                              α                i                                                        ,                          ⁢                          ⁢                                    ∂            L                                ∂            b                          =                  0          =                                    >                              s                j                                      =                                                            β                  j                                +                                  u                  j                                                            c                2                                                        ,                          ⁢                          ⁢                                    ∂            L                                ∂            b                          =                  0          =                                    >                              r                j                                      =                                                            γ                  j                                +                                  v                  j                                                            c                2                                                        ,      
It can be obtained with substitution:
            min              ω        ,        b        ,        ξ        ,        r        ,        s              ⁢          (                                    1            2                    ⁢                                    ∑                                                i                  1                                =                1                            l                        ⁢                                                  ⁢                                          ∑                                                      i                    2                                    =                  1                                l                            ⁢                                                          ⁢                                                y                                      i                    1                                                  ⁢                                  y                                      i                    2                                                  ⁢                                  α                                      i                    1                                                  ⁢                                  α                                      i                    2                                                  ⁢                                  K                  ⁡                                      (                                                                  x                                                  i                          1                                                                    ,                                              x                                                  i                          2                                                                                      )                                                                                      +                              1                          2              ⁢                                                          ⁢                              c                1                                              ⁢                                    ∑                              i                =                1                            l                        ⁢                                                  ⁢                          α              i              2                                      +                              1            2                    ⁢                                    (                                                                    ∑                                          i                      =                      1                                        l                                    ⁢                                                                          ⁢                                                            y                      i                                        ⁢                                          α                      i                                                                      +                                                      ∑                                          j                      =                                              l                        +                        1                                                                                    l                      +                      k                                                        ⁢                                                                          ⁢                                      (                                                                  β                        j                                            -                                              γ                        j                                                              )                                                              )                        2                          +                              1            2                    ⁢                                    ∑                                                j                  1                                =                                  l                  +                  1                                                            l                +                k                                      ⁢                                                  ⁢                                          ∑                                                      j                    2                                    =                                      l                    +                    1                                                                    l                  +                  k                                            ⁢                                                          ⁢                                                (                                                            β                                              j                        1                                                              -                                          γ                                              j                        1                                                                              )                                ⁢                                  (                                                            β                                              j                        2                                                              -                                          γ                                              j                        2                                                                              )                                ⁢                                  K                  ⁡                                      (                                                                  x                                                  j                          1                                                                    ,                                              x                                                  j                          2                                                                                      )                                                                                      +                              ∑                          i              =              1                        l                    ⁢                                          ⁢                                    ∑                              j                =                                  l                  +                  1                                                            l                +                k                                      ⁢                                                  ⁢                                          y                i                            ⁢                                                α                  i                                ⁡                                  (                                                            β                      j                                        -                                          γ                      i                                                        )                                            ⁢                              K                ⁡                                  (                                                            x                      i                                        ,                                          x                      j                                                        )                                                                    +                              1                          c              2                                ⁢                                    ∑                              j                =                                  1                  +                  k                                                            l                +                k                                      ⁢                                                  ⁢                                          (                                                      β                    j                                    +                                      u                    j                                                  )                            ⁢                              (                                                      γ                    j                                    +                                      v                    j                                                  )                                                    -                              ∑                          i              =              1                        l                    ⁢                                          ⁢                      α            i                          -                              ∑                          j              =                              1                +                k                                                    l              +              k                                ⁢                                          ⁢                      (                                          β                j                            +                              γ                j                                      )                              )                          ⁢                  s        .        t        .                                  ⁢                  α          i                    ,              β        j            ,              γ        j            ,              u        j            ,                        v          j                ≥        0.            